The present invention relates essentially to a computer for numeric calculation of a plurality of functionally interrelated data for a plurality of points, forming a grid in rows and columns, of a domain with preassigned initial or boundary conditions, by iterative interpolation on the values of adjacent points of the grid. Such computers may be used, for example, to approximate solutions of an ordinary or partial differential equation or system of such differential equations. Moreover, the computers may be employed for solving other problems in which a plurality of functionally interrelated data are to be calculated, such as so-called Monte Carlo problems, problems in economics, and the like.
Heretofore, there have been no analytical methods for solving differential equations, and, more particularly, partial differential equations, for boundary conditions and initial values of any complexity. It has been necessary, therefore, to resort to numerical approximation, based generally on the idea of converting a differential equation into difference equations. Geometrically, this amounts to replacing the tangent at the coordinate point by the secant. It may also be interpreted as replacing a Taylor expansion at the coordinate point by its first term.
When systems comprising a number of equations are to be solved, exact methods are not available because the unavoidable rounding off errors will build up and the numerical solution will diverge. In practice, systems comprising a number of differential equations are often solved by iterative methods, i.e., methods of "relaxation", such as the Jacobi whole-step method, the Gauss-Seidel single-step method, and the step-by-step hyperrelaxation method. Disadvantages of these methods are their slow convergence and consequently, long computer time. Even calculating the solutions for the grid points of a 50 .times. 50-point matrix of a domain will require several hours of computer time on fast computers for a differential equation of even moderately complicated structure.
In natural science and engineering there are a number of problems which depend almost entirely on numerical solutions of differential equations. Examples include problems involving potential and wave equations in electrical engineering, stress analysis in machine design and construction work, the Navier-Stokes equations in fluid transfer, equations for transfer of matter in process engineering, heat conduction and diffusion in thermodynamics, and long-term, wide-area weather forecasting in meteorology. In the field of physics, furthermore, a sub-branch has evolved that likewise employs the above-mentioned methods among others, and has been termed "computational physics".
The reasons for the comparatively long computation times involved in the solution of such systems of equations lie in the universality of electronic computers. Accordingly, computers have been incapable, heretofore, of achieving rapid, simultaneous calculation of a considerable number of data units related to each other by some functional relationship and, thus, could not solve, within a satisfactory time, the above-identified problems.